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Links
A quoted post from mathematician @prof_g argues that AI is expanding mathematical idea-space faster than people can follow, making it look negatively curved (hyperbolic) rather than clustered around famous conjectures. The upshot, he warns, is lots left to discover but researchers drifting apart and working alone.
- Mathematicians have traditionally clustered around famous conjectures, which the author calls locally positively curved idea-space.
- AI is pushing the boundaries outward quickly, revealing the space as closer to hyperbolic, with more territory the further you go.
- The result: even small separations between people or agents could mean wandering alone, with fewer shared discoveries.
Mitchell Zemil, a freelance animator and adjunct professor, writes in response to 3blue1brown's AI-in-math discussion, arguing that artists and mathematicians both value the human labor, intentionality, and personal "DNA" behind their work. He contends that generative AI turns making into tastemaking, and that mathematics is more than an answer-producing machine.
- Art is valued for the artist's labor, skill development, and intentional process, so generative AI's ability to produce polished output quickly threatens that value and raises ethical issues around training data.
- Past technologies like cameras and digital tools democratized art without erasing the human practice behind it, but generative AI is different because it shifts the work toward selecting from outputs rather than making them.
- Mathematics is often treated as a purely objective pursuit that machines could finish off, but Zemil argues people still value living mathematicians who explain, interpret, and push the frontier, much as audiences still value dancers at the ballet.
The New York Times profiles Terence Tao, the mathematician who became a tenured professor at 24, and why he has ended up as the field's unofficial spokesman on AI. The piece argues that established academics like him feel less threatened by AI than younger researchers just starting out.
- Tao says senior academics are not as threatened by AI as people beginning their careers.
- His celebrity has made him the de facto public voice for mathematics on AI, and that role now takes up most of his time.
- The math community is looking to him for answers, which the headline frames as a "war footing."
Researchers used BootLoops, an AI tool built with Claude, to scan scientific literature across 22 fields and identify problems that could be solved by borrowing techniques from unrelated disciplines. In 3 months, the team produced 36 papers solving problems ranging from calculating evolutionary tree probabilities to creating an 8x larger database of language sound systems.
- BootLoops discovered that many unsolved problems in different fields share identical mathematical structures with known solutions from physics, computer science, or math—but researchers didn't know to look across disciplines.
- The AI solved 15 previously unsolved integrals in high-energy physics and created ACCSTACK, a database of 6,000+ languages' stress and tone patterns that's eight times larger than the previous largest dataset.
- The approach requires heavy human judgment to work: the AI lacks "research taste" and can't determine what scientists actually care about, and it tends to favor older, highly cited work over new problems.
Steven Byrnes argues that LLMs perform well at math not because math is easy to verify, but because mathematical literature is almost entirely correct—giving the model clean training data to learn from. The "verifiability" explanation is circular since LLMs themselves do the verifying.
- The common claim that "math is easy to verify" doesn't explain LLM math ability, because verification still relies on the LLM's judgment, which is what we're trying to explain in the first place.
- Mathematical literature is >99% accurate, so imitative learning on this data naturally produces correct mathematical reasoning without needing heavy reinforcement learning or formal verification systems.
- The theory predicts which domains LLMs should excel at: those with high-quality, accurate training data (like math and coding) versus those with noisier training signals (like subjective judgments in fiction or social domains).
This MIT course focuses on estimating answers and tackling problems through intuitive shortcuts instead of formal proofs or exact calculations. It covers guesswork techniques and heuristic strategies to quickly arrive at useful results.
- MIT's Street Fighting Mathematics (IAP 2008, taught by Scott Aaronson, J. Michael Steele, Steven G. Johnson) teaches estimation and heuristic problem-solving instead of rigorous proofs.
- Key techniques include dimensional analysis, bounding arguments, perturbation methods, and picking simpler functions to bound nasty integrals.
- The course derives results like the √N law for random walks and Stirling's formula through physical intuition and rough approximation rather than formal derivation.
- These heuristics extend beyond physics to algorithm analysis, offering quick mental shortcuts for estimating runtime or complexity (e.g., spotting O(N log N) behavior without formal analysis).
Information theory is a mathematical framework for quantifying, storing, and communicating information, established by Claude Shannon in the 1940s. It involves concepts such as entropy, mutual information, and channel capacity, with applications ranging from data compression to cryptography and artificial intelligence. The field combines insights from various disciplines including mathematics, engineering, and computer science.
- Shannon's 1948 work founded information theory by formalizing entropy as a measure of uncertainty/information content
- Channel capacity defines the theoretical maximum rate for reliable data transmission over a noisy channel
- The framework underpins practical technologies like data compression and error-correcting codes, extending into cryptography and AI